### Conditional supply-link chance

We decide the conditional supply-link chance *p*(*s*|*c*) by evaluating the agency communication community, proven in Fig. 1d, with ground-truth data on the true customer-supplier relations, obtained from a nation-wide survey among the many members of a giant enterprise illustration group that represents corporations throughout all sectors besides agriculture, carried out in April 2020. Within the on-line survey greater than 100,000 corporations and companies have been requested to share their ten most important suppliers and prospects, respectively. Greater than 5,900 corporations declared not less than one provider or buyer with a complete of greater than 17,000 customer-supplier relations reported. For particulars on the survey, see SI Textual content 2. We acquire the general chance {that a} provide hyperlink exists between two corporations, provided that they’d not less than one dialog occasion within the noticed time interval of roughly 125 days, is (p(s|c) = 0.19). For the conditional communication chance we get (p(c|s) = 0.27). For comparability, the respective marginal chance from the agency communication community instantly is (p(c)=0.002). For the linking chance –using Hungarian data– we get (p(s)=0.00005). Each values are orders of magnitude smaller than the conditional chances, indicating extremely vital hyperlink correlations between the availability and communication layers.

The conditional hyperlink chance will increase with the depth of the firm-firm communication. As a proxy for the latter we use the common day by day name period, ({bar{d}}_{ij}), in seconds per day. In Fig. 2a *p*(*s*|*c*) is proven conditional that ({bar{d}}_{ij}) is bigger than a threshold ({bar{d}}) (purple). The variety of hyperlinks used to calculate the overlap decreases as a perform of the edge ({bar{d}}) and is proven in blue. *p*(*s*|*c*) rises from 19% to values round 70% for ({bar{d}}_{ij}>30)s/d and round 90% for 60s/d. The variety of hyperlinks reduces from 75 to 14 hyperlinks as ({bar{d}}) will increase. Notice that errors don’t improve, as a result of a better chance is related to a smaller error. For particulars of the computation and errorbars, see SI Textual content 2.

For the availability community (*p*(*c*|*s*)) the perfect proxy for tie power can be the quantity of traded items. Nevertheless, this data is just not accessible. Following the philosophy of “gravity fashions” in economics, we assume that giant and small corporations sometimes commerce giant and small volumes, respectively^{44}. The gravity mannequin implies that the hyperlink weight is proportional to the product of the agency’s sizes. Right here, to remain constant on the communication knowledge, we proxy the agency dimension with the variety of gadgets related to a agency. Supplementary Fig. S3 exhibits *p*(*c*|*s*) for the networks thresholded by the variety of gadgets per agency in purple and the variety of hyperlinks within the underlying pattern in blue. We discover a rise from 27% to round 60% for the community of corporations with 4 or extra gadgets. For thresholds bigger than 4, the curve ranges off and stabilizes round 70% for thresholds of 6 or extra gadgets. The variety of hyperlinks drops as in Fig. 2a, however once more, the error-bars are nonetheless small enough.

### Reconstructing the availability community

To acquire an estimate of the availability community, based mostly on the communication community, c, we assume all communication hyperlinks, (c_{ij}), with a name period of greater than 30 seconds per day, ({bar{d}}_{ij}=30)s/d, to sign a provide relation, (s_{ij}). With this threshold we intention to stability the lack of data resulting from ignored provide hyperlinks and growing hyperlink correlations as a result of thresholds. This specific threshold is the results of a minimization of the Kullback-Leibler divergence for diploma distributions of the Hungarian provide community (HSN) and thresholded FCNs, described in SI Textual content 3. We arrive at an unweighted and undirected reconstructed provide community (RSN). To get an estimate for the hyperlink instructions (agency *i* provides *j,* or vice versa), we use classical input-output tables of the nationwide statistical workplace. They include data on the quantity of commerce between financial sectors within the economic system. A component of the input-output desk, (G_{ab}), describes the circulate of products (in Euro) from sector *a* to sector *b*. We denote the variety of hyperlinks (firm-firm provide relations) from sector *a* to sector *b* by (L_{ab}) and assume that the ratio of hyperlinks from one sector to the opposite is proportional to the ratio of products flowing between these sectors, ({L_{ab}}/{L_{ba}} approx {G_{ab}}/{G_{ba}}). For instance, the circulate between the agricultural sector (*a*) and the meals trade (*f*) is (G_{af} approx 3,400 mathrm {m})€, whereas the meals trade offered items for (G_{fa} approx 450 mathrm {m})€ to the agricultural sector. We now assume that it’s({3,400}/{450} approx 7.6) occasions extra seemingly {that a} provide hyperlink factors from a agency *a* to 1 in *f*. We now contemplate each hyperlink from agency *i* in sector *a* to agency *j* in sector *b* within the RSN and assign it a path in response to the chance

$$start{aligned} p(i rightarrow j) = frac{G_{ab}}{G_{ab}+G_{ba}} , . finish{aligned}$$

(1)

Since we carry out this project stochastically, we should always assume in ensembles of RSNs. Lastly, we estimate a supply-link weight for each hyperlink within the RSN. We use the businesses’ whole property, calculated from the stability sheets, as dimension data, (s_i); it’s obtained from a commercially accessible enterprise intelligence database, see Supplies and Strategies. As earlier than, within the spirit of “gravity fashions”^{44}, we estimate the hyperlink weight between corporations *i* and *j* proportional to the product of agency sizes, (W_{ij} = s_i s_j). We are going to use solely relative weights within the following.

### Evaluating community topologies of supply-chains, firm-firm communication, and human communication

It’s enlightening to match the community topology of the so-obtained RSN (blue in Fig. 2b) with the topologies of the Hungarian provide community (HSN) (orange) (for which the actual topology is thought^{18}) and the personal communication community between particular person individuals (inexperienced) (i.e. not between corporations). Determine 2b exhibits the diploma distribution of the RSN (blue) compared to the precise HSN derived from VAT knowledge^{31}. Each networks are related and fats tailed, in distinction to the human communication community (HCN) that was obtained from the cell phone knowledge set. The RSN has a mean diploma of (langle ok^{RSN} rangle = 4.79). Its diploma distribution has a most at (ok^{RSN}=2) and its fats tail will be approximated by an influence regulation exponent (alpha _k^{RSN} = 2.18(12)) for (ok^{RSN}>30). The HSN doesn’t present a rise for small *ok* but in addition displays a fats tail with (alpha _k^{HSN} = 2.40(3)), for (ok^{HSN}>30). The typical diploma is (langle ok^{HSN} rangle = 2.1). For the HCN we discover a mean diploma of (langle ok^{HCN} rangle = 4.75). There, the lower of *p*(*ok*) for prime values is stronger, with an exponent of (alpha _k^{HCN} = 4.89(26)) for (ok^{HCN}>20). For a extra detailed comparability of community traits, together with the clustering coefficient and nearest neighbor levels, see SI Textual content 4, SI Fig. S5 and SI Tab. S1.

### Financial systemic threat

With an affordable reconstruction of the availability community, RSN, we flip to the quantification of financial systemic threat within the nationwide manufacturing community. For this we use the financial systemic threat index (ESRI) as developed in^{18}. The underlying precept of the ESRI algorithm is sketched in Fig. 3a, in an instance with seven corporations of equal dimension throughout the identical industrial sector. The ESRI of agency *i*, assumes that if agency *i* (purple cross) can’t function for a while (e.g. defaults) it neither provides nor calls for inputs. The purchasers and suppliers of *i* then cut back their manufacturing accordingly, inflicting a successive discount of manufacturing of their prospects and suppliers. This recursive discount converges to a state the place all corporations have lowered their stage of manufacturing in response to *i*’s default. Determine 3a exhibits the relative discount for each agency. The fraction of whole financial exercise misplaced is the (ESRI_i) of agency *i*. We use the ESRI with a heuristically calibrated generalized Leontief manufacturing perform (SI Eq. (9)), which captures the main variations between corporations with bodily manufacturing capabilities (e.g. agriculture, manufacturing, and so forth., i.e. NACE A01-F43) and corporations producing companies (NACE G45-U99). For the primary group, bodily inputs (i.e. merchandise equipped by corporations from NACE A01-F43) are important to their manufacturing course of and, consequently, their missing causes main disruptions in agency’s manufacturing in a non-linear vogue. In financial phrases, these are Leontief inputs. Nevertheless, service inputs reminiscent of consulting companies, journey company companies and so forth. should not important for the bodily manufacturing and disrupt manufacturing solely in a linear manner. For corporations producing companies we assume that every one inputs disrupt their manufacturing in a linear vogue. For an in depth clarification of using generalized Leontief manufacturing capabilities within the ESRI definition, we consult with SI Textual content 5 and the “Generalized Leontief” state of affairs in^{18}.

We compute the ESRI for each agency within the community and plot their values in response to their rank, from highest ESRI to lowest, in Fig. 3b. That is known as the *systemic threat profile* of the manufacturing community. The ESRI for the defaulting agency in panel a is highlighted because the purple bar. Performing the identical steps for all corporations in a single realization of the RSN yields the systemic threat profile proven in Fig. 3c, the place we present the 200 riskiest corporations. The profile exhibits related traits to what has been reported for the precise manufacturing community of Hungary^{18}, particularly, a plateau containing the 65 most dangerous corporations, which all, apart from just a few extraordinarily dangerous corporations, have an identical threat of round (mathrm {ESRI} approx 0.21), adopted by a pointy decline for corporations that aren’t a part of the plateau. The inset in Fig. 3c exhibits the cumulative distribution (CDF) (p(mathrm {ESRI} > x)) of the ESRI in log-log scale.

To take the stochastic nature of the RSN under consideration we repeat the ESRI calculation. We contemplate 5 realizations of the RSN to calculate their imply ESRI. Subsequently, resulting from computational challenges, we deal with the 1000 most dangerous corporations solely, after rating them in response to their imply ESRI. For these we repeat the ESRI calculation 100 occasions. For every node we get a distribution of ESRI values. Determine 3d exhibits the median ESRI for each agency as a strong line; the 25% and 75% quantiles are indicated by the errorbars. Another solution to examine the ESRI profile of the RSN is to plot the maximal systemic threat of each node. This methodology yields related outcomes and is proven in SI Fig. S6 in SI Textual content 6. The median ESRI per node profile in Fig. 3d exhibits the identical traits as the one run in Fig. 3c, a plateau of high-risk corporations and a speedy decline of ESRI exterior of the plateau. In distinction to the one run ESRI profile, the plateau consists of solely round 50 corporations. The unfold of the ESRI distributions for particular person nodes is small for high- in addition to low-risk nodes, indicating that the outcomes are remarkably steady and sturdy. For the intermediate threat corporations error-bars develop into giant, indicating that their ESRI depends upon the path of 1 or few hyperlinks. It’s a well-known function of systemic threat and the ESRI that single hyperlinks or hyperlink instructions can have a big affect^{11,18}. Ref.^{18} explains that some nodes “inherit” systemic threat by being a vital provider to a agency that’s inherently dangerous resulting from e.g. its dimension. Due to this fact flipping a link-direction can flip a node from a vital provider of a central agency to a purchaser of that agency, which strongly reduces its inherited systemic threat.

To grasp which corporations are within the plateau of Fig. 3c, in Fig. 4, we plot the firm-size, approximated by the corporations’ whole property towards the ESRI. It’s evident that the excessive systemic threat plateau (highlighted in purple) accommodates giant and small corporations, with their whole property spanning greater than 4 orders of magnitude. Though agency dimension correlates nicely with ESRI (Spearman’s (rho =0.87)), it isn’t predictor for systemic threat, since for a given firm-size the ESRI can range by a number of orders of magnitude. An analogous state of affairs is described in^{18}.

The 65 corporations discovered within the excessive systemic threat plateau primarily belong to the manufacturing sector (NACE lvl. 1 class C, 77%), adopted by corporations within the electrical energy, gasoline stream and air-con provide (D, 8%) and monetary and insurance coverage actions (Okay, 6%) sectors. The complete composition contains 22 NACE lvl. 2 sectors and is listed in SI Tab. S2. In distinction to the precise Hungarian manufacturing community^{18}, a number of corporations from non-manufacturing sectors (NACE (ge) 45) are discovered within the plateau. That is considerably sudden since they’re related to linear manufacturing capabilities (see SI Textual content 5), which causes their shock spreading conduct to be much less excessive than for Leontief producers.

### Robustness of outcomes

Our research is topic to a number of limitations, particularly (i) the imperfect overlap of the 2 communication and supply-link layers, limiting the doable accuracy, (ii) the restricted market protection of the cellphone supplier (leading to restricted settlement even when (p(s|c) = 1)), see SI Textual content 7, and (iii) errors originating from the community reconstruction uncertainties within the estimations of instructions and weights.

To estimate the biases and errors launched by these weaknesses, we carry out a number of simulation research. First, we generate an artificial communication community based mostly on the HSN and the possibilities to discover a communication hyperlink, the place a supply-link is current *p*(*c*|*s*), and the place no supply-link is discovered (p(c|lnot s)). From this artificial communication community we then take a pattern of nodes in response to an estimated market share *m* of the information supplier and calculate the induced subgraph comprised by hyperlinks solely between the sampled nodes.

Lastly, following the process used on the empirical knowledge, we reconstruct a provide community from this artificial communication community and calculate the ESRI. We calculate Spearman’s rank correlation coefficient, (rho), between the ESRI as calculated on the total, actual HSN and on the reconstructed subgraph. After repeating these steps for 100 occasions with (m=1/3), (p(c|s) = 0.21) and (p(c|lnot s) = 9.3 occasions 10^{-5}), we discover a mean Spearman correlation of (langle rho (ESRI_{HSN}, ESRI_{reconstr})rangle = 0.563(6)). In SI Textual content 7 we handle the shortcomings talked about above one after the other and focus on the anticipated magnitude of the launched errors. We discover that probably the most related impact is brought on by the restricted market share with a drop of correlation of (Delta langle rho rangle = 0.31), adopted by the restricted overlap, including one other, (Delta langle rho rangle = 0.13). The results from community reconstruction cut back the correlation by solely (Delta langle rho rangle = 0.0004), which is remarkably small. We calculate the chance {that a} node that’s among the many 0.1% riskiest nodes of the subsample can be among the many riskiest 0.1% of *all* nodes and discover 32.9(82)%. The chance that one of many high 0.1% of the subsample nodes is among the many high 1% of the total community is 47.7(99)%.